WINO: A Weak-Form Physics Informed Neural Operator for Hyperelasticity on Variable Domains

2026-05-23

AI Summary

This work addresses the challenge of efficiently solving hyperelasticity problems on domains with varying geometry by proposing a novel approach that eliminates the need for body-fitted meshes and reference solution data. The method uniquely embeds weak-form physics into a neural operator, integrating φ-finite element methods (φ-FEM) with level-set-based geometric representation. Training is performed in a data-free manner through weak-form residuals and penalty terms from auxiliary equations, while uniformly handling both Dirichlet and Neumann boundary conditions. Furthermore, a Neural Operator Warm-Start (NOWS) strategy is introduced to significantly accelerate the convergence of nonlinear φ-FEM iterations. Numerical experiments demonstrate that the proposed framework achieves errors below 0.04 across all benchmark cases and reduces total computational time by 50%–80% compared to purely data-driven approaches.

Abstract

We propose a Weak-form Physics-Informed Neural Operator (WINO), a data-free framework that combines the efficiency of neural operators with the geometric flexibility of the $\varphi$-finite element method ($\varphi$-FEM). $\varphi$-FEM is an unfitted method that accommodates geometric variations without body-fitted meshes, where the domain geometry is represented by the level-set function $\varphi$. To impose the boundary conditions, Dirichlet problems adopt the $\varphi$-FEM lifting so only the homogeneous displacement contribution is learned, whereas traction-driven Neumann problems additionally predict the auxiliary fields necessary for the unfitted weak formulation. Parameters are trained by minimizing squared weak-form residuals aligned with $\varphi$-FEM together with squared penalties on the cut-cell auxiliary equations, which removes the need for large paired datasets of converged reference solutions. After training, WINO outputs can seed the nonlinear $\varphi$-FEM solvers as neural operator warm starts (NOWS), which reduce iteration counts relative to traditional cold-started solvers. Numerical benchmarks show that WINO achieves high accuracy below 0.04 across all benchmarks, while reducing total computational time by 50--80\% compared with purely data-driven methods.

Problem

• hyperelasticity
• variable domains
• physics-informed neural operator
• weak-form
• boundary conditions

Innovation

• Physics-Informed Neural Operator
• Weak-form formulation
• φ-FEM
• Variable domains
• Data-free learning